Axiom:Robinson Axioms

Axioms

Robinson's axioms are a set of axioms for the language of arithmetic.

They are weaker than Peano's Axioms by design, and are intended to be the minimal amount of logic required to carry out proofs such as Godel's Incompleteness Theorems.

$(\text Q 1)$	$:$	$\ds \forall x, y:$	$\ds \map s x = \map s y \implies x = y $
$(\text Q 2)$	$:$	$\ds \forall x:$	$\ds \map s x \ne 0 $
$(\text Q 3)$	$:$	$\ds \forall x:$	$\ds x \ne 0 \implies \exists y: x = \map s y $
$(\text Q 4)$	$:$	$\ds \forall x:$	$\ds x + 0 = x $
$(\text Q 5)$	$:$	$\ds \forall x, y:$	$\ds x + \map s y = \map s {x + y} $
$(\text Q 6)$	$:$	$\ds \forall x:$	$\ds x \times 0 = 0 $
$(\text Q 7)$	$:$	$\ds \forall x, y:$	$\ds x \times \map s y = \paren {x \times y} + x $

This entry was named for Raphael Mitchel Robinson.

\((\text Q 1)\)	$:$	\(\ds \forall x, y:\)	\(\ds \map s x = \map s y \implies x = y \)
\((\text Q 2)\)	$:$	\(\ds \forall x:\)	\(\ds \map s x \ne 0 \)
\((\text Q 3)\)	$:$	\(\ds \forall x:\)	\(\ds x \ne 0 \implies \exists y: x = \map s y \)
\((\text Q 4)\)	$:$	\(\ds \forall x:\)	\(\ds x + 0 = x \)
\((\text Q 5)\)	$:$	\(\ds \forall x, y:\)	\(\ds x + \map s y = \map s {x + y} \)
\((\text Q 6)\)	$:$	\(\ds \forall x:\)	\(\ds x \times 0 = 0 \)
\((\text Q 7)\)	$:$	\(\ds \forall x, y:\)	\(\ds x \times \map s y = \paren {x \times y} + x \)